Diffie Helman, Successive Square, Modulo Arithmetic
Explained
eCommerce couldn’t exist without the ability for two computers to share secret
information. When eCommerce first began, their was a pervasive concern that credit
cards would be compromised, that buying online wouldn’t be “safe.”
The secret to keeping things secret is encryption. And the secret to encryption is having a
shared secret, something that only the sender and receiver know. It’s the “key.” In the
world of espionage spies could meet at the secret meeting place and exchange a code
book, but that’s hard to do when it’s you and Amazon. No place to meet.
Enter the Diffie-Helman Key Exchange Protocol. This piece of math magic lets two
people communicate in the clear (or, in plain English – two computer send messages that
are available to everyone), and at the end of the exchange, the two computers know a
secret key that others can’t figure out.
There are three parts to this bit of magic: modular arithmetic, raising numbers to a power
with successive squaring, and last, but not least, the Diffie-Helman exchange.
Part 1: Modular Arithmetic.
It’s not as complicated as it seems. In modular arithmetic, numbers are never bigger than
the “modulus.” For example, for “mod 12” numbers are never bigger than 11. You can
think of a clock. There’s no 13 or 14.
The easiest way to convert numbers in modular arithmetic is by dividing at taking the
remainder. For those of you who are calculator slaves you may have forgotten what
remainders are. It’s the integers left over after you divide. Take 17 / 3. That’s 5 with a
remainder of 2 (because 3 goes into 18 5 times – 3*5 =15 – and there are 2 left over). On
a calculator you get 5.6666666. The remainder is NOT .66666.
One more example.
435 mod 17 is 10 because 435 / 17 = 25 with 10 left over.
A word about notation. When we say “mod 17” that means that the whole expression is
modular, not just the last number.
There’s one more critical piece to modular arithmetic. You can do the reduction at
whenever the numbers are bigger than the modulus. Again, this concept is simpler than it
sounds.
We’ll do some examples.
12 * 29 * 408 mod 13
Remember that the phrase “mod 13” applies to the whole expression. We can reduce the
parts before we do the multiplication. The is VERY important as we shall soon see.
12 mod 13 = 12
29 mod 13 = 3
408 mod 13 = 5
Now do the multiplication. 12 * 3 = 36, but “mod 13” it is 10
Then 10 * 5 = 50, but “mod 13” that’s 11 – which is our answer.
Just for fun, we can do the mod reduction at some other point.
12*29 = 348 348*408 = 141,984
141,984 mod 13 = 11
See – same result. That’s not a proof at all, but it serves to demonstrate the point.
The beauty of modular arithmetic is that the numbers stay manageable.
Part 2: Successive Squaring
Let’s say I want to calculate 3^9 (3 raised to the 9th power). That’s
3*3*3*3*3*3*3*3*3.
It’s the same as 3^7 8 3, which is the same as 3^4 * 3^4 * 3, or even 3^2 * 3^2 *3^2 *
3^2 *3 . Remember, we add exponents when we multiply. Either one of these has 9 3’s
multiplied together.
That’s all there is to successive squares.
Let’s do the example:
3^1
3
3^2
9
3^4
81
3^8
6561
3^9 = 3^8 * 3 = 6561*3 = 19,683
We only had to do 4 multiplications, not 9.
Consider what happens if the exponents get bigger. What it we wanted to do, say,
3^4098?
For starters, the numbers are enormous. 3^32 is 1,853,020,188,851,840. And we’d need
to do 5831 multiplications. With successive squaring though, twelve multiplications get
us to 3^4096, then one more and we’re there (3^4096 * 3^2).
It’s time to combine Part 1 and Part 2. The problem with raising things to big powers is
that the numbers get huge fast. Remember the tale of the beggar who asked the king to
put one grain of sand on the first square of the chessboard, two on the next, four and on
the third. The universe doesn’t have 2^64 grains of sand.
Let’s do 3^4098 say, mod 11. Now the number won’t ever get big.
3^1 = 3 mod 11 = 3
3^2 = 9 mod 11 = 9
3 ^ 4 = 81 mod 11 = 4
OK. Stop right here. I’m not going to do 81 * 81. We already know that 81 mod 11 = 4.
SO – 3^8 = 4 * 4 = 16 mod 11 = 5
3^16 = 5*5 = 25 mod 11 = 3
3^32 …..you’ve got the picture
3^4096 mod 11 = 3
3^4098 = 3 * 9 = 27 mod 11 = 5! And we only had to do 13 multiplications.
Part 3: Diffie Helman
This really does seem like magic. Two people talk, everyone listens, and in the end, they
share a secret number.
Here’s how it works. We start with a number, “g” that everyone in the whole world
knows.
Next, you and I each pick a secret number. Call mine “a” and yours “b.”
I’ll calculate g^a and tell you. Let’s call that “A.” You calculate g^b and tell me. We’ll
call that “B.”
Finally, I’ll calculate B^a and you calculate A^b. Presto - we have a secret number
because:
A = g^a so A^b = (g^a)^b = g^(a*b).
And
B = g^b so B^a = (g^b)^a = g^(b*a)
What makes it really secret is that we do all of this calculation using modular arithmetic.
If we all know what g is, and I tell you what A is, you can’t possibly figure out a. Trust
me on that one.
That’s all there is to it.